Abstract
We prove that every Hilbert space operator which factorizes invariantly through Sobolev space \(W_1^1({{\mathbb{T}}^d})\) belongs to some non-trivial Schatten class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Blei, The Grothendieck inequality revisited, Memoirs of the American Mathematical Society 232 (2014).
M. Braverman, K. Makarychev, Y. Makarychev and A. Naor, The Grothendieck Constant is strictly smaller than Krivine’s bound, in 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science—FOCS 2011, IEEE Computer Society, Los Alamitos, CA, 2011, pp. 453–462.
J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press, Cambridge, 1995.
D. J. H. Garling, Inequalities, Cambridge University Press, Cambridge, 2007.
A. Grothendieck, Résuméde la théorie métrique des produits tensoriels topologiques, Boletim da Sociedade de Matemática de São Paulo 8 (1953), 1–79.
L. Hörmander, Estimates for translation invariant operators in Lpspaces, Acta Matheatica 104 (1960), 93–140.
S. V. Kisljakov, Sobolev imbedding operators, and the nonisomorphism of certain Banach spaces, Funkcional’nyi Analis i ego Priloženija 9 (1975), 22–27.
J. Lindenstrauss and A. Pelczynski, Absolutely summing operators in Lp-spaces and their applications, Studia Mathematica 29 (1968), 275–326.
F. Lust-Piquard and Q. Xu, The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators, Journal of Functional Analysis 244 (2007), 488–503.
A. Naor and O. Regev, Krivine schemes are optimal, Proceedings of the American Mathematical Society 142 (2014), 4315–4320.
A. Pelczyński and M. Wojciechowski, Paley projections on anisotropic Sobolev spaces on tori, Proceedings of the London Mathematical Society 65 (1992), 405–422.
G. Pisier, Grothendieck’s theorem, past and present, Bulletin of the American Mathematical Society 49 (2012), 237–323.
M. Wojciechowski, On the summing property of the Sobolev embedding operators, Positivity 1 (1997), 165–169.
P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, Vol. 25, Cambridge University Press, Cambridge, 1991.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kazaniecki, K., Pakosz, P. & Wojciechowski, M. An invariant version of the little Grothendieck theorem for Sobolev spaces. Isr. J. Math. 244, 33–47 (2021). https://doi.org/10.1007/s11856-021-2166-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-021-2166-5